Riemann theta functions
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1: 21.2 Definitions
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§21.2(i) Riemann Theta Functions
… ►For numerical purposes we use the scaled Riemann theta function , defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. … ►§21.2(ii) Riemann Theta Functions with Characteristics
… ►§21.2(iii) Relation to Classical Theta Functions
…2: 21.8 Abelian Functions
§21.8 Abelian Functions
… ►For every Abelian function, there is a positive integer , such that the Abelian function can be expressed as a ratio of linear combinations of products with factors of Riemann theta functions with characteristics that share a common period lattice. …3: 21.10 Methods of Computation
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§21.10(i) General Riemann Theta Functions
… ►§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
… ►Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.
4: 21.9 Integrable Equations
§21.9 Integrable Equations
… ►Typical examples of such equations are the Korteweg–de Vries equation … ► … ►
5: 21.1 Special Notation
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►Uppercase boldface letters are real or complex matrices.
►The main functions treated in this chapter are the Riemann theta functions
, and the Riemann theta functions with characteristics .
►The function
is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
6: 21.3 Symmetry and Quasi-Periodicity
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§21.3(i) Riemann Theta Functions
… ► ►§21.3(ii) Riemann Theta Functions with Characteristics
… ► …For Riemann theta functions with half-period characteristics, …7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
8: 21.4 Graphics
§21.4 Graphics
►Figure 21.4.1 provides surfaces of the scaled Riemann theta function , with … ►For the scaled Riemann theta functions depicted in Figures 21.4.2–21.4.5 … ►9: 21.5 Modular Transformations
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§21.5(i) Riemann Theta Functions
… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which is determinate: … ►§21.5(ii) Riemann Theta Functions with Characteristics
… ►For explicit results in the case , see §20.7(viii).10: 21.7 Riemann Surfaces
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